Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission

We present a theoretical study of a new application of a simple pi-phase-shifted fiber Bragg grating (PSFBG) in transmission mode as a high-speed optical temporal integrator. The PSFBG consists of two concatenated identical uniform FBGs with a pi phase shift between them. When the reflectivities of the FBGs are extremely close to 100%, the transmissive PSFBG can perform the time integral of the complex envelope of an arbitrary input optical signal with high accuracy. As an example, the integrator is numerically shown to be able to convert an input Gaussian pulse into an optical step signal.     

Image encryption using fractional Mellin transform,structured phase filters,and phase retrieval

An image encryption scheme has been presented by using two structured phase masks in the fractional Mellin transform (FrMT) plane of a system, employing a phase retrieval technique. Since FrMT is a non-linear integral transform, its use enhances the system security. We also add further security features by carrying out spatial filtering in the frequency domain by using a combination of two phase masks: a toroidal zone plate (TZP) and a radial Hilbert mask (RHM). These masks together increase the key space making the system more secure. The phase key used in decryption has been obtained by applying an iterative phase retrieval algorithm based on the fractional Fourier transform. The algorithm uses amplitude constraints of secret target image and the ciphertext (encrypted image) obtained from multiplication of fractional Mellin transformed arbitrary input image and the two phase masks (TZP and RHM). The proposed encryption scheme has been validated for a few grayscale images, by numerical simulations. The efficacy of the scheme has been evaluated by computing mean-squared-error (MSE) between the secret target image and the decrypted image. The sensitivity analysis of the decryption process to variations in various encryption parameters has also been carried out.     

Image displacement based on phase-encoded reference joint fractional transform correlator

In order to utilize the space of an input plane efficiently and make the optical structure more flexible, an image displacement measurement based on phase-encoded reference joint fractional transform correlator (PER-JFrTC) is proposed. We use a random phase mask to encode the reference image and overlay it with the target image forming the input image. Joint power spectrum (JPS) of the input image is obtained by Fourier transform and the resultant is encoded by the same phase mask. Then a fractional Fourier transform with an order p is applied to the phase-encoded JPS (PJPS), resulting in a correlation output with a sharp cross-correlation peak, which includes the displacement information between the reference and the target image. Contrast to displacement measurement based on traditional joint transform correlator (JTC), PER-JFrTC can use the space of the input plane efficiently and reduces the influence of the auto-correlation. Also the position of cross-correlation peak can be fixed arbitrarily according to the fractional order p as well as the optical set-up can be more flexible and easier to implement. Results based on digital computation show that PER-JFrTC could detect the displacement accurately and verify our proposal. A possible optical set-up is suggested.     

一阶光学系统分数傅里叶变换的相空间分析

在维格纳相空间中,通过将一阶光学系统的传输矩阵分解为坐标旋转、比例缩放和啁啾矩阵的组合,得到了一阶光学系统在空域的分数傅里叶表示.结果表明:任意一阶光学系统均可表示为经过比例缩放和二次相位调制的分数傅里叶变换.通过将输入输出光场在相空间中作π/2角旋转,得到了一阶光学系统在频域的传输矩阵和衍射积分公式,进而得到了一阶光学系统在频域的分数傅里叶表示.比较空域和频域一阶光学系统的相空间变换矩阵,说明2个系统本质上属同一变换在不同基坐标下的表示,并推导出了光学系统在空域和频域具有相同分数傅里叶变换的条件.     

Fractional Fourier plane image encryption technique using radial hilbert-, and Jigsaw transform

A new method for image encryption using integral order radial Hilbert transform (RHT) filter in the fractional Fourier transform (FRT) domain has been proposed. The technique is implemented using the popular double random phase encoding method in the fractional Fourier domain. The random phase masks (RPMs), integral orders of the RHT, fractional orders of FRT, and indices of the Jigsaw transform (JT) have been used as keys for encryption and decryption. Simulation results have been presented and the schematic representation for optical implementation has been proposed. The mean-square-error and signal-to-noise ratio between the decrypted image and the input image have been calculated for the correct as well as incorrect orders of the RHT. Effect of occlusion and noise on the performance of the proposed scheme has also been studied. The robustness of the technique has been verified against attack using partial windows of the correct random phase masks. Similar investigations have also been carried out for the chosen-, and the known-plain-text attacks.     

分数傅里叶变换计算全息

在计算全息和分数傅里叶变换的基础上提出了不对称分数傅里叶变换计算全息和双随机相位不对称分数傅里叶变换计算全息。在这种方法中,首先用一随机相位函数乘以输入图像信息,然后沿x方向实施α级次的一维分数傅里叶变换,再乘以第二个随机相位函数,最后,沿y方向实施β级次的一维分数傅里叶变换。采用迂回位相编码法对变换后的结果编码,绘出计算全息图。为了恢复原始图像,需要知道变换级次和随机相位函数。利用这种方法进行图像加密,使加密图像的密钥由原来两重增加到四重,从而提高了系统的保密性能。     

Terahertz bandwidth all-optical Hilbert transformers based on long-period gratings

A novel, all-optical design for implementing terahertz (THz) bandwidth real-time Hilbert transformers is proposed and numerically demonstrated. An all-optical Hilbert transformer can be implemented using a uniform-period long-period grating (LPG) with a properly designed amplitude-only grating apodization profile, incorporating a single π-phase shift in the middle of the grating length. The designed LPG-based Hilbert transformers can be practically implemented using either fiber-optic or integrated-waveguide technologies. As a generalization, photonic fractional Hilbert transformers are also designed based on the same optical platform. In this general case, the resulting LPGs have multiple π-phase shifts along the grating length. Our numerical simulations confirm that all-optical Hilbert transformers capable of processing arbitrary optical signals with bandwidths well in the THz range can be implemented using feasible fiber/waveguide LPG designs.     

Zernike-basis expansion of the fractional and radial Hilbert phase masks

The linear Hilbert phase mask or transform has found applications in image processing and spectroscopy. An optical version of the fractional Hilbert mask is considered here, comprising an imaging system with a circular, unobscured pupil in which a variable phase delay is introduced into one half of the pupil, split bilaterally.The radial Hilbert phase mask is also used in image processing and to produce optical vortices which have applications in optical tweezers and the detection of exoplanets.We subjected the fractional and radial Hilbert phase masks to Zernike function expansion in order to compute the image plane electromagnetic field distribution using Nijboer-Zernike theory. The Zernike functions form an orthogonal basis on the unit circle. The complex-valued Zernike expansion coefficients for these two phase masks were derived for use in the context of the Extended Nijboer-Zernike (ENZ) theory of image formation. The ENZ approach is of interest in that it allows a greater range of defocus to be dealt with, provides a simple means of taking a finite source size into account and has been adapted to high Numerical Aperture (NA) imaging applications.Our image plane results for the fractional Hilbert mask were verified against a numerical model implemented in the commercial optical design and analysis code, Zemax^®. It was found that the Nijboer-Zernike result converged to the Zemax^® result from below as the number of Zernike terms in the expansion was increased.     

Optical image encryption using fractional Fourier transform and chaos

We propose a new method for image encryption using fractional Fourier transform and chaos theory. Random phase masks are generated using iterative chaos functions. The input image is combined with the first random phase mask at the object plane and is then transformed using the fractional Fourier transform. After the first fractional Fourier transform, the second random phase mask, again generated by using the chaos functions, is used at the fractional plane. The second fractional Fourier transform operation is then carried out to obtain the encrypted image. Three types of chaos functions have been used: the logistic map, the tent map and the Kaplan–Yorke map. The mean square error and the signal-to-noise ratio between the decrypted image and the input image for the correct order and the incorrect order of the fractional Fourier transform have been calculated. The computer simulations are presented to verify the validity of the proposed technique.     

可变可转单波片偏振控制器研究

基于PLZT波片的相对相位延迟量可变性和其折射率椭球主光轴的可转性,提出后反馈单个PLZT可变可转波片偏振控制器.采用模拟退火算法进行扰动寻优,仿真结果显示这种控制器是有效的：能将任意输入偏振态转换为稳定的线偏振光而其光强达到总光强的98%以上,在控制过程中相移能在限定范围内变化,避免了由复位重置可能引起的波动.     

Bragg光纤光栅傅里叶模式耦合理论

建立了Bragg光纤光栅傅里叶模式耦合理论.在分析光纤光栅的耦合模时,发现了耦合模式的振幅系数间存在傅里叶变换关系.推导了将傅里叶变换和模式耦合融合在一起的Bragg光纤光栅反射谱和透射谱的通用表达式.该理论是用傅里叶变换得到Bragg光纤光栅折射率微扰的空域谱,再对该空域谱进行模式耦合分析计算,从而得到Bragg光纤光栅的光谱特性.根据该理论,仿真分析了Bragg光纤光栅的谱特性,与耦合模理论、直接傅里叶变换法进行了对比分析.结果表明,傅里叶模式耦合理论与传统的耦合模理论及实际Bragg光纤光栅的光谱特性一致,具有简单、清晰、直接、精确和分析效率高的特点,可分析任意轴向折射率微扰分布的Bragg光纤光栅结构.     

长周期光纤光栅傅里叶模式耦合理论

建立了长周期光纤光栅傅里叶模式耦合理论.在分析同向模式耦合时,发现了同向耦合模式的振幅系数间存在傅里叶变换关系.推导了长周期光纤光栅的同向耦合谱和透射谱的通用表达式.该理论是用傅里叶变换分析得出长周期光纤光栅折射率微扰的空域谱,再对该空域谱进行模式同向耦合分析,从而得到长周期光纤光栅光谱特性的通用表达式.根据该理论模拟分析了长周期光纤光栅在不同长度和微扰幅值时的光谱特性,与传统耦合模理论进行了对比分析.结果表明,该长周期光纤光栅傅里叶模式耦合理论具有简单、精确和高效的特点,与实际长周期光纤光栅的透射谱特性一致.应用该理论可分析无过耦合的任意轴向折射率微扰分布的长周期光纤光栅光谱特性.     

Proposal and design of an in-fiber all-optical fractional integrator

We theoretically and numerically demonstrate that a single fiber Bragg grating – conveniently apodized and of uniform period – operated in reflection can perform an arbitrary-order fractional integration of an input optical waveform. Analytical expressions were found relating the fractional integration order with the apodization profile of the fiber Bragg grating. This simple device shows a good accuracy calculating the fractional time integral of the complex field of arbitrary input optical waveforms.     

Image encryption based on gyrator transform and two-step phase-shifting interferometry

A novel optical image encryption method is proposed, based on gyrator transform and phase-shifting interferometry. The input two-dimensional image to be encrypted is gyrator transformed two times, and two random phase masks are placed at the input plane and the output plane of the first gyrator transform. Two-step phase-shifting interferometry is used to record the digital holograms of the input image encrypted by use of double-random phase encoding technique in gyrator transform domain. The rotation angles of gyrator transform, the random phase mask in the gyrator plane and the arbitrary phase shift used for recording form the keys for decryption of the input image. Numerical simulations are presented to verify its validity and efficiency.     

一种基于希尔伯特变换实施相移的相位解调算法

相移法因其精度高、运算简单,所以被广泛应用于形貌检测中的相位解调。该方法的弱点是:不仅要采集多幅相移图像,而且不能克服相移器的移相误差。将小波分析与希尔伯特变换结合起来,采用小波分析去除直流背景成分,利用希尔伯特变换方法实现90°相移;利用相移法解调形貌相位,由于是采用软件来实现相移的,因此只需采集一幅变形栅线图。对该方法与传统相移技术进行了模拟比较实验,结果表明,该方法具有较高的精度。     

Chirped fiber Bragg gratings with arbitrary group delay responses using stepped-stretching fiber method

We present the design and development of piece-wise stepped-chirp fiber Bragg gratings (FBGs) with arbitrary group delay responses using a uniform phase mask in a pre-stretched fiber. The method involves writing a series of sub-gratings on a pre-stretched fiber whose length is varied during the UV exposure process. Two motorized stages are employed to control the length of the pre-stretched fiber in order to adjust the grating pitch at each writing step to achieve piece-wise stepped chirping of the FBG. To demonstrate the effectiveness of the method to produce chirped FBGs with arbitrary group delay responses, a linear-chirp FBG and a quadratic-chirp FBG were fabricated, and their measured performances agree well with analytical predictions. PACS 42.81.-i; 42.79.Dj; 42.79.Ci     

Generation of dark hollow beams by using a fractional radial Hilbert transform system

The radial Hilbert transform has been extend to the fractional field, which could be called the fractional radial Hilbert transform (FRHT). Using edge-enhancement characteristics of this transform, we convert a Gaussian light beam into a variety of dark hollow beams (DHBs). Based on the fact that a hard-edged aperture can be expanded approximately as a finite sum of complex Gaussian functions, the analytical expression of a Gaussian beam passing through a FRHT system has been derived. As a numerical example, the properties of the DHBs with different fractional orders are illustrated graphically. The calculation results obtained by use of the analytical method and the integral method are also compared.     

Hilbert reconstruction of phase-shifted second-harmonic holographic images

New techniques are presented that make phase-shifting holography viable for second-harmonic generation (SHG) holography with weak object fields. We developed an intrinsic phase shift calibration of SHG holograms, an algorithm that extracts the reference and object intensity directly from a set of phase-shifted holographic data, and a more robust phase-shifting holography reconstruction algorithm based on π-shifted hologram pairs that permits self-calibration of the phase shift and recovery of the complex field through a Hilbert transform.     

Necessity analyses of phase masks in joint fractional Fourier transform correlator

In this paper, the mathematical expressions and the particularities of the joint fractional Fourier transform correlator (JFRTC) are detailedly analyzed. In a JFRTC, the distance between the two fractional correlation peaks can be adjusted flexibly by shifting the fractional orders. However, the intensity of the fractional correlation peaks decreases rapidly as the first fractional order p₁ drops. The former property is very useful in a practical recognition system, but the latter one seriously reduces the recognition sensitivity of a JFRTC. For this reason, phase masks should be added at the input plane of the JFRTC to enhance the intensities of the fractional correlation peaks and effectively overcome the default. Compared with the existing JFRTC with single phase modulation, a JFRTC with double phase modulation is suggested to more flexibly improve the performance of a JFRTC.     

Analytical expression of pulse broadening in an arbitrary linear slow light medium

We derive an analytical expression for the broadening of a Gaussian input pulse in an arbitrary linear slow light medium. The expression consists of two terms, one corresponding to amplitude broadening (low-pass filtering of the pulse bandwidth) and another corresponding to phase broadening (phase dispersion around the resonance). It is shown that for a Lorentzian gain profile, the amplitude broadening is dominant at small fractional delays. However, for large fractional delays, phase broadening is inevitably dominate.